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Decision rule - Wikipedia
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Function that maps an observation to an action
In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game theory.
In order to evaluate the usefulness of a decision rule, it is necessary to have a loss function detailing the outcome of each action under different states.
Given an observable random variable X over the probability space ( X , Σ , P θ ) {\displaystyle \scriptstyle ({\mathcal {X}},\Sigma ,P_{\theta })} , determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ : X {\displaystyle \scriptstyle {\mathcal {X}}} → A.
Examples of decision rules[edit]
- An estimator is a decision rule used for estimating a parameter. In this case the set of actions is the parameter space, and a loss function details the cost of the discrepancy between the true value of the parameter and the estimated value. For example, in a linear model with a single scalar parameter θ {\displaystyle \theta } , the domain of θ {\displaystyle \theta } may extend over R {\displaystyle {\mathcal {R}}} (all real numbers). An associated decision rule for estimating θ {\displaystyle \theta } from some observed data might be, "choose the value of the θ {\displaystyle \theta } , say θ ^ {\displaystyle {\hat {\theta }}} , that minimizes the sum of squared error between some observed responses, and responses predicted from the corresponding covariates given that you chose θ ^ {\displaystyle {\hat {\theta }}} ." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined, θ ^ {\displaystyle {\hat {\theta }}} could be chosen, for instance, using some optimization algorithm.
- Out of sample prediction in regression and classification models.
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